Appendix S1 Stability of the zooplankton–fish system given in eqns. (1).
In addition to the non-trivial solution given by eqns. (3) and (4), there are two trivial
solutions with regard to distribution strategies: (α1, α2; β1, β2) = (1, 0; 1, 0) and (0, 1; 0,
1). The first one corresponds to the situation that all zooplankton and fish stay in the
pelagic area, and the second one that all in the littoral area. With trivial solutions, the
total abundances of zooplankton and fish are:
 s  c   1 (q1mg  b1 )d1  q1mg  b1 
,
( Z 10 , F10 )   1  1

s1  when (α1, α2; β1, β2) = (1,
2
e



1
1
1

 

s
0; 1, 0), and ( Z 01 , F01 )   2
 e
 c 2   2 (q 2mg  b2 )d 2  q 2mg  b2 

,

s 2  when (α1,
2
 22
 2


α2; β1, β2) = (0, 1; 0, 1).
Here I extend the concept of evolutionary stable strategies (ESS, Maynard Smith
1982) to distribution stability analysis of the model. Invasion by small numbers of
mutants of zooplankton (Z') and fish (F') with different distribution strategies (αi' and
βi') into the system with ESS (αi* and βi*) at the non-trivial equilibrium abundances
( Z and F ) is described by the following dynamics:
 'Z'
 ' Z '   * F  ' F ' 
dZ ' i 
  si (qimg  bi ) i   i i  i  i 
dt
si
si  si
si  

i
    i i ' Z '  i ' F ' / si 
(A.1)
1
 'F'
dF ' i    i* Z  i ' Z '   i ' F '

  si e i 

 ci i
dt
si  si
si
  si
i ' F '
 i
i
si
  i* F  i ' F '   i ' F ' 

 d i 


s
s
i
 i
 si 


  e i i ' Z '  i ' F ' / si  d i (  i ' F ' ) 2 / si .
(A.2)
Combining eqns. (A.1) and (A.2), I obtain


i
d
(eZ ' F ' )    d i (  i ' F ' ) 2 / si .
dt
(A.3)
Provided that either d1 or d2 is positive, eqns. (A.1) and (A.3) lead to Z' → 0 and
eZ'+F' → 0 with t → ∞, indicating F' → 0. This means that mutants of zooplankton
(Z') or fish (F') cannot invade into the system and that the distribution strategies αi*
and βi* for zooplankton and fish, respectively, are ESS.
Invasion by small numbers of mutants Z' and F' with different distribution
strategies (αi' and βi') into the system with a trivial solution (e.g. α1 = β1 = 1, α2 = β2 =
0, Z 10 , F10 ) is given by
i

 F  'Z' 2
dZ '
  si  (qimg  bi )   i i 10  i
 o (Z ' , F ' )
dt
si  si

 b1 1 ' Z '  0
(A.4)
2
i

 Z
 F   ' F' 2
dF '
  si  e i i 10  ci   i  d i i 10  i
 o (Z ' , F ' )
dt
si
si  si

 (c2   2 )  ' F '  0,
(A.5)
where I neglected the second-order infinitesimal terms o2(Z', F') between '  ' signs. A
mutant population of zooplankton can invade the system (eqn. A.4), and therefore this
trivial solution is not ESS. Similarly, the other trivial solution (i.e., α1 = β1 = 0, α2 = β2
= 1, Z 01 , F01 ) is not ESS, either.
As for the stability of population dynamics with the non-trivial solution, let me
consider the following function:
Z
F
Z
F
V ( Z , F )  eZ   1  log   F   1  log  ,
Z
F
Z
F
(A.6)
Since the inequality x  1  log x  0 holds for any given variable x  0 except for
x  1, the function V ( Z , F )  0 where the equality holds at the equilibrium
abundances ( Z , F ) . Differentiating the function with regard to time, I have
i
dV V dZ V dF
2


 F  F   d i  i2 / si  0 ,
dt
Z dt F dt
(A.7)
where the last equality holds at the equilibrium abundances, too. Thus the function V
monotonically decreases with time when di > 0, converging to the minimum at
3
( Z , F ) , and is a so-called Lyapunov function of the system (La Salle & Lefschetz
1961). In other words, the equilibrium abundances ( Z , F ) are globally stable. A
positive di was assumed because fish in lakes are likely to have their territories for
breeding, food and predation avoidance in the littoral habitat covered with
macrophytes.
Cressman et al. (2004) showed that systems with predator–prey interactions in a
two-habitat environment always lead to the same population equilibria and their
corresponding distributions. On the other hand, they also showed that systems with
two competing species may not lead to the same predictions on population and
distribution stability.
REFERENCES
Cressman, R., Křivan, V. & Garay, J. (2004). Ideal free distributions, evolutionary
games, and population dynamics in multiple-species environments. Am. Nat.,
164, 473–489.
La Salle, J. & Lefschetz, S. (1961). Stability by Liapunov's direct method, with
applications. Academic Press, New York.
Maynard Smith, J. (1982). Evolution and the theory of games. Cambridge University
Press, Cambridge.
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